[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.65,0:00:02.70,Default,,0000,0000,0000,,We're asked to solve\Nthe quadratic equation,
Dialogue: 0,0:00:02.70,0:00:07.62,Default,,0000,0000,0000,,negative 3x squared plus\N10x minus 3 is equal to 0.
Dialogue: 0,0:00:07.62,0:00:09.56,Default,,0000,0000,0000,,And it's already written\Nin standard form.
Dialogue: 0,0:00:09.56,0:00:11.06,Default,,0000,0000,0000,,And there's many\Nways to solve this.
Dialogue: 0,0:00:11.06,0:00:13.60,Default,,0000,0000,0000,,But in particular, all solve\Nit using the quadratic formula.
Dialogue: 0,0:00:13.60,0:00:14.68,Default,,0000,0000,0000,,So let me just rewrite it.
Dialogue: 0,0:00:14.68,0:00:19.44,Default,,0000,0000,0000,,We have negative 3x squared\Nplus 10x minus 3 is equal to 0.
Dialogue: 0,0:00:19.44,0:00:20.81,Default,,0000,0000,0000,,And actually, I'll\Nsolve it twice
Dialogue: 0,0:00:20.81,0:00:22.72,Default,,0000,0000,0000,,using the quadratic\Nformula to show you
Dialogue: 0,0:00:22.72,0:00:25.35,Default,,0000,0000,0000,,that as long as we manipulated\Nthis in the valid way,
Dialogue: 0,0:00:25.35,0:00:26.85,Default,,0000,0000,0000,,the quadratic\Nformula will give us
Dialogue: 0,0:00:26.85,0:00:29.90,Default,,0000,0000,0000,,the exact same roots or\Nthe exact same solutions
Dialogue: 0,0:00:29.90,0:00:31.37,Default,,0000,0000,0000,,to this equation.
Dialogue: 0,0:00:31.37,0:00:34.32,Default,,0000,0000,0000,,So in this form right over\Nhere, what are our ABCs?
Dialogue: 0,0:00:34.32,0:00:36.78,Default,,0000,0000,0000,,Let's just remind ourselves\Nwhat the quadratic formula even
Dialogue: 0,0:00:36.78,0:00:37.45,Default,,0000,0000,0000,,is actually.
Dialogue: 0,0:00:37.45,0:00:38.73,Default,,0000,0000,0000,,That's a good place to start.
Dialogue: 0,0:00:38.73,0:00:40.64,Default,,0000,0000,0000,,The quadratic formula\Ntells us that if we
Dialogue: 0,0:00:40.64,0:00:43.17,Default,,0000,0000,0000,,have a quadratic\Nequation in the form ax
Dialogue: 0,0:00:43.17,0:00:48.32,Default,,0000,0000,0000,,squared plus bx plus c is equal\Nto 0, so in standard form,
Dialogue: 0,0:00:48.32,0:00:52.36,Default,,0000,0000,0000,,then the roots of this are\Nx are equal to negative b
Dialogue: 0,0:00:52.36,0:00:55.63,Default,,0000,0000,0000,,plus or minus the\Nsquare root of b
Dialogue: 0,0:00:55.63,0:01:02.43,Default,,0000,0000,0000,,squared minus 4ac,\Nall of that over 2a.
Dialogue: 0,0:01:02.43,0:01:05.24,Default,,0000,0000,0000,,And this is derived from\Ncompleting the square
Dialogue: 0,0:01:05.24,0:01:06.41,Default,,0000,0000,0000,,in a general way.
Dialogue: 0,0:01:06.41,0:01:09.91,Default,,0000,0000,0000,,So it's no magic here, and I've\Nderived it in other videos.
Dialogue: 0,0:01:09.91,0:01:11.33,Default,,0000,0000,0000,,But this is the\Nquadratic formula.
Dialogue: 0,0:01:11.33,0:01:12.91,Default,,0000,0000,0000,,This is actually giving\Nyou two solutions,
Dialogue: 0,0:01:12.91,0:01:14.89,Default,,0000,0000,0000,,because you have the\Npositive square root here
Dialogue: 0,0:01:14.89,0:01:16.68,Default,,0000,0000,0000,,and the negative square root.
Dialogue: 0,0:01:16.68,0:01:19.75,Default,,0000,0000,0000,,So let's apply it here in the\Ncase where-- in this case,
Dialogue: 0,0:01:19.75,0:01:26.89,Default,,0000,0000,0000,,a is equal to negative\N3, b is equal to 10,
Dialogue: 0,0:01:26.89,0:01:31.00,Default,,0000,0000,0000,,and c is equal to negative 3.
Dialogue: 0,0:01:31.00,0:01:33.24,Default,,0000,0000,0000,,So applying the quadratic\Nformula right here,
Dialogue: 0,0:01:33.24,0:01:36.53,Default,,0000,0000,0000,,we get our solutions to be\Nx is equal to negative b.
Dialogue: 0,0:01:36.53,0:01:37.53,Default,,0000,0000,0000,,b is 10.
Dialogue: 0,0:01:37.53,0:01:42.80,Default,,0000,0000,0000,,So negative b is negative 10\Nplus or minus the square root
Dialogue: 0,0:01:42.80,0:01:44.69,Default,,0000,0000,0000,,of b squared.
Dialogue: 0,0:01:44.69,0:01:45.46,Default,,0000,0000,0000,,b is 10.
Dialogue: 0,0:01:45.46,0:01:50.65,Default,,0000,0000,0000,,So b squared is 100\Nminus 4 times a times c.
Dialogue: 0,0:01:50.65,0:01:54.39,Default,,0000,0000,0000,,So minus 4 times negative\N3 times negative 3.
Dialogue: 0,0:01:54.39,0:01:55.47,Default,,0000,0000,0000,,Let me just write it down.
Dialogue: 0,0:01:55.47,0:01:58.93,Default,,0000,0000,0000,,Minus 4 times negative\N3 times negative 3.
Dialogue: 0,0:01:58.93,0:02:01.10,Default,,0000,0000,0000,,All of that's under\Nthe radical sign.
Dialogue: 0,0:02:01.10,0:02:02.86,Default,,0000,0000,0000,,And then all of that is over 2a.
Dialogue: 0,0:02:02.86,0:02:05.54,Default,,0000,0000,0000,,So 2 times a is negative 6.
Dialogue: 0,0:02:05.54,0:02:07.66,Default,,0000,0000,0000,,So this is going to be\Nequal to negative 10
Dialogue: 0,0:02:07.66,0:02:14.71,Default,,0000,0000,0000,,plus or minus the square root\Nof 100 minus-- negative 3 times
Dialogue: 0,0:02:14.71,0:02:16.08,Default,,0000,0000,0000,,negative 3 is positive 9.
Dialogue: 0,0:02:16.08,0:02:18.28,Default,,0000,0000,0000,,Positive 9 times\N4 is positive 36.
Dialogue: 0,0:02:18.28,0:02:19.53,Default,,0000,0000,0000,,We have a minus sign out here.
Dialogue: 0,0:02:19.53,0:02:21.61,Default,,0000,0000,0000,,So minus 36.
Dialogue: 0,0:02:21.61,0:02:23.86,Default,,0000,0000,0000,,All of that over negative 6.
Dialogue: 0,0:02:23.86,0:02:27.31,Default,,0000,0000,0000,,This is equal to\N100 minus 36 is 64.
Dialogue: 0,0:02:27.31,0:02:31.83,Default,,0000,0000,0000,,So negative 10 plus or\Nminus the square root of 64.
Dialogue: 0,0:02:31.83,0:02:33.90,Default,,0000,0000,0000,,All of that over negative 6.
Dialogue: 0,0:02:33.90,0:02:35.94,Default,,0000,0000,0000,,The principal square\Nroot of 64 is 8.
Dialogue: 0,0:02:35.94,0:02:38.23,Default,,0000,0000,0000,,But we're taking the positive\Nand negative square root.
Dialogue: 0,0:02:38.23,0:02:43.94,Default,,0000,0000,0000,,So this is negative 10 plus\Nor minus 8 over negative 6.
Dialogue: 0,0:02:43.94,0:02:45.62,Default,,0000,0000,0000,,So if we take the\Npositive version,
Dialogue: 0,0:02:45.62,0:02:48.21,Default,,0000,0000,0000,,we say x could be\Nequal to-- negative 10
Dialogue: 0,0:02:48.21,0:02:52.56,Default,,0000,0000,0000,,plus 8 is negative\N2 over negative 6.
Dialogue: 0,0:02:52.56,0:02:54.57,Default,,0000,0000,0000,,So that was taking\Nthe plus version.
Dialogue: 0,0:02:54.57,0:02:56.33,Default,,0000,0000,0000,,That's this right over here.
Dialogue: 0,0:02:56.33,0:02:58.94,Default,,0000,0000,0000,,And negative 2 over\Nnegative 6 is equal to 1/3.
Dialogue: 0,0:02:58.94,0:03:00.84,Default,,0000,0000,0000,,If we take the\Nnegative square root,
Dialogue: 0,0:03:00.84,0:03:04.61,Default,,0000,0000,0000,,negative 10 minus 8-- So let's\Ntake negative 10 minus 8.
Dialogue: 0,0:03:04.61,0:03:08.03,Default,,0000,0000,0000,,That would be x is equal\Nto-- negative 10 minus 8
Dialogue: 0,0:03:08.03,0:03:10.07,Default,,0000,0000,0000,,is negative 18.
Dialogue: 0,0:03:10.07,0:03:13.49,Default,,0000,0000,0000,,And that's going to\Nbe over negative 6.
Dialogue: 0,0:03:13.49,0:03:16.93,Default,,0000,0000,0000,,Negative 18 divided by\Nnegative 6 is positive 3.
Dialogue: 0,0:03:16.93,0:03:19.38,Default,,0000,0000,0000,,So the two roots for\Nthis quadratic equation
Dialogue: 0,0:03:19.38,0:03:22.49,Default,,0000,0000,0000,,are positive 1/3 and positive 3.
Dialogue: 0,0:03:22.49,0:03:24.70,Default,,0000,0000,0000,,And I want to show you the\Nwe'll get the same answer,
Dialogue: 0,0:03:24.70,0:03:25.74,Default,,0000,0000,0000,,even if we manipulate this.
Dialogue: 0,0:03:25.74,0:03:27.28,Default,,0000,0000,0000,,Some people might\Nnot like the fact
Dialogue: 0,0:03:27.28,0:03:30.12,Default,,0000,0000,0000,,that our first coefficient\Nhere is a negative 3.
Dialogue: 0,0:03:30.12,0:03:31.88,Default,,0000,0000,0000,,Maybe they want a positive 3.
Dialogue: 0,0:03:31.88,0:03:33.25,Default,,0000,0000,0000,,So to get rid of\Nthat negative 3,
Dialogue: 0,0:03:33.25,0:03:37.01,Default,,0000,0000,0000,,they can multiply both sides of\Nthis equation times negative 1.
Dialogue: 0,0:03:37.01,0:03:39.27,Default,,0000,0000,0000,,And then if you did\Nthat, you would get 3x
Dialogue: 0,0:03:39.27,0:03:45.00,Default,,0000,0000,0000,,squared minus 10x plus 3 is\Nequal to 0 times negative 1,
Dialogue: 0,0:03:45.00,0:03:46.90,Default,,0000,0000,0000,,which is still equal to 0.
Dialogue: 0,0:03:46.90,0:03:51.99,Default,,0000,0000,0000,,So in this case, a is equal to\N3, b is equal to negative 10,
Dialogue: 0,0:03:51.99,0:03:54.43,Default,,0000,0000,0000,,and c is equal to 3 again.
Dialogue: 0,0:03:54.43,0:03:56.40,Default,,0000,0000,0000,,And we could apply\Nthe quadratic formula.
Dialogue: 0,0:03:56.40,0:04:00.56,Default,,0000,0000,0000,,We get x is equal to\Nnegative b. b is negative 10.
Dialogue: 0,0:04:00.56,0:04:02.46,Default,,0000,0000,0000,,So negative negative\N10 is positive
Dialogue: 0,0:04:02.46,0:04:05.30,Default,,0000,0000,0000,,10, plus or minus\Nthe square root
Dialogue: 0,0:04:05.30,0:04:07.95,Default,,0000,0000,0000,,of b squared, which is\Nnegative 10 squared,
Dialogue: 0,0:04:07.95,0:04:12.11,Default,,0000,0000,0000,,which is 100, minus\N4 times a times c.
Dialogue: 0,0:04:12.11,0:04:16.07,Default,,0000,0000,0000,,a times c is 9 times 4 is 36.
Dialogue: 0,0:04:16.07,0:04:17.98,Default,,0000,0000,0000,,So minus 36.
Dialogue: 0,0:04:17.98,0:04:20.25,Default,,0000,0000,0000,,All of that over 2 times a.
Dialogue: 0,0:04:20.25,0:04:21.99,Default,,0000,0000,0000,,All of that over 6.
Dialogue: 0,0:04:21.99,0:04:28.03,Default,,0000,0000,0000,,So this is equal to 10 plus or\Nminus the square root of 64,
Dialogue: 0,0:04:28.03,0:04:30.66,Default,,0000,0000,0000,,or really that's\Njust going to be 8.
Dialogue: 0,0:04:30.66,0:04:32.21,Default,,0000,0000,0000,,All of that over 6.
Dialogue: 0,0:04:32.21,0:04:36.42,Default,,0000,0000,0000,,If we add 8 here, we get\N10 plus 8 is 18 over 6.
Dialogue: 0,0:04:36.42,0:04:38.31,Default,,0000,0000,0000,,We get x could be equal to 3.
Dialogue: 0,0:04:38.31,0:04:41.30,Default,,0000,0000,0000,,Or if we take the negative\Nsquare root or the negative 8
Dialogue: 0,0:04:41.30,0:04:43.10,Default,,0000,0000,0000,,here, 10 minus 8 is 2.
Dialogue: 0,0:04:43.10,0:04:46.28,Default,,0000,0000,0000,,2 over 6 is 1/3.
Dialogue: 0,0:04:46.28,0:04:50.30,Default,,0000,0000,0000,,So once again, you get\Nthe exact same solutions.